A current drawn on a page has sources

Seven rungs of this anchor project a scalar field — a height, a temperature, an illumination — and ask what the page does to its gradient. Everything measured there is about direction and steepness, and the answer has been that a conformal map keeps the direction and gets the size wrong.

The other half of what gets mapped is a vector field: a wind, an ocean current, a plate velocity, a migration. A scalar field has a gradient; a vector field has a divergence and a curl, and those are not statements about direction at all. They are statements about area.

What is being conserved, and what is not #

A flow of an incompressible quantity has zero divergence: what goes into a region comes out of it, and there are no sources or sinks. That is a physical statement about the ground and it is the first thing anybody reads off a flow map — a place where the arrows converge is a place where something is piling up.

Draw the same flow on a page and take the divergence of what is drawn. The answer is not zero unless the map preserves area.

The reason is short. Divergence is flux per unit area, and a map that changes areas changes the denominator. A current running from ground the map draws small into ground the map draws large is drawn spreading out, and spreading out is exactly what a source looks like.

The conformal maps failing is the part worth stopping on. Water runs downhill on the ground, not on the page establishes that a conformal projection is the right one for a scalar field, because it keeps every direction and therefore keeps the path a drop of water takes. Point it at a vector field and it keeps every direction and destroys the conservation law, which is the property the picture is usually being read for. The two halves of the same anchor want opposite projections.

Why the conformal maps fail too #

It is worth being explicit about the case a reader is most likely to get wrong, because the reasoning that gets it wrong is otherwise sound.

Conformality is a statement about angles at a point. The arrows on a conformal map point exactly where the current points on the ground, at every point, with no error at all — that is what the property means and this collection has spent a field establishing it. So a reader tracing a streamline on Mercator is tracing the true streamline, exactly.

Divergence is not made of angles. It is flux over area, and a conformal map’s areas are wrong by $\sec^2\varphi$ — so the streamlines are right and the spacing between them is not. That is the whole of it: a set of correct streamlines drawn at incorrect spacings, and the spacing is what carries the conservation.

The picture is right about where the water goes and wrong about how much of it there is, which is a genuinely awkward combination because the two are read off the same drawing by the same glance.

The closed form #

The invented divergence has an exact expression, and it is short enough to be useful.

$$\nabla_{\text{page}} \cdot \mathbf{v}_{\text{page}} \;=\; \mathbf{v} \cdot \nabla \ln s$$

where $s$ is the areal scale factor. The apparent source strength is the component of the flow along the gradient of the log areal factor: the flow moving into ground the map is enlarging.

Two consequences follow immediately. An equal-area map has $\nabla \ln s = 0$ everywhere, so the invented divergence is zero for any flow whatsoever — the theorem holds for every field, not for the one measured here. And a flow running along a contour of the areal factor picks up nothing even on a badly distorting map, so the failure is directional: the same map is honest about a current running east along a parallel and dishonest about the same current running north.

The size, and where it is #

Across a hundred-kilometre box the invented divergence reaches 5.5 per cent of the local flow on the plate carrée, 3.9 on Miller and 3.0 on Mercator, with means around one per cent. Those are worst cases over the sphere, and the worst case is the honest number to quote for a defect that concentrates: a mean of one per cent spread evenly would be ignorable, and a mean of one per cent that is five per cent in one band and nothing elsewhere is a five per cent problem in that band.

The plate carrée being the worst is not an accident either, and it repeats a finding from two anchors along. The average was a choice of norm finds the equirectangular last of ten on scale departure at high exponents — excellent nearly everywhere and catastrophic in a small place — and the same shape produces the same answer here, because the gradient of the areal factor is exactly the quantity that is large where a map’s behaviour changes fastest. Five per cent is not a rounding error in a flow map: it is the size of the convergence zones a reader is looking for.

And the effect grows with latitude, because $\nabla \ln s$ does. On Mercator the log areal factor is $2\ln\sec\varphi$ and its gradient is $2\tan\varphi/R$, so the invented divergence doubles between 45° and 63° and doubles again by 74°. A flow map is least trustworthy exactly where the polar circulations are, which is where an oceanographic or atmospheric flow map is most often about something.

The Lambert cylindrical’s flat line at zero across the whole range is worth reading as more than a control. It is the same projection that the average was a choice of norm finds unremarkable on every distortion criterion and that nobody chooses for anything — and for this one purpose it is not merely better than the alternatives but exactly right, at every latitude, with no residual to argue about. A property that is either perfectly preserved or not preserved at all does not produce a ranking; it produces a shortlist.

What it does to a reading #

The practical shape of the damage is worth stating in the terms a reader of a flow map actually uses.

A convergence zone on a current map is a place where arrows crowd together, and it is read as somewhere water is piling up or sinking. A divergence is read as upwelling. On an equal-area map both readings are sound. On Mercator, at sixty degrees, a purely meridional current picks up an apparent divergence of about three per cent of itself across a hundred kilometres — which is the same order as the real convergences at an ocean front, and which has the same visual signature.

The two are separable in principle: the invented part is a known function of position and the projection, so it can be subtracted. It is separable in practice only by somebody who knows the projection, has the areal factor and thinks to do the arithmetic, which is not the reader the map was drawn for.

What makes it worse than an ordinary bias is the sign structure. Every invented source is paid for by an invented sink somewhere else, and they integrate to zero over the sphere — so no check on the total flux, and no check on the global mass balance, gives them away. The picture conserves everything globally and lies locally, which is exactly the arrangement under which a defect survives every audit anybody would think to run.

What was computed, and how #

The flow is constructed to be divergence-free rather than tested for it: it is the perpendicular of the exact analytic gradient of a stream function, which makes its divergence zero by an identity rather than by a cancellation. The gate still measures it on the sphere, and it comes out at 1.6 × 10⁻⁹, which is the derivative’s own noise and is the floor everything else is judged against.

The drawn field at a point is the image of the ground velocity under the projection’s own local frame — the page displacement per metre east and per metre north, which is the same matrix the composition rung works in. That is what a flow map is: an arrow whose direction and length are the image of a ground velocity.

The page divergence is then taken in page coordinates, by stepping in $x$ and $y$ on the page and differencing, which is what a reader’s eye or a program handed the drawing computes. It is not the sphere divergence converted; that would be assuming the answer.

The assertions require five things separately: that the flow be divergence-free on the sphere at the noise floor; that every equal-area projection return a page divergence at the same floor; that every projection that is not equal-area return one that is not, including the conformal ones; that the measurement follow $\mathbf{v}\cdot\nabla\ln s$ everywhere; and that the size be a material share of the flow across an ordinary box, since an effect below a per cent would be a curiosity.

The projection that would be right #

Since the requirement is exact and narrow, it is worth asking what an honest flow map would be drawn on.

Any equal-area projection works, and the choice among them is then free for other reasons — which is an unusually comfortable position for this collection to arrive at, since every other choice it prices is a trade. There is no trade here: the property is preserved by a whole family, and within that family a designer can optimise shape, or the region, or legibility, with no cost to the conservation law at all.

The awkwardness is that flow maps are conventionally drawn on Mercator, for a reason that is itself sound: Mercator exists because a constant bearing is a straight line, and a current chart is a navigational document. So the convention that makes a chart useful for steering is the convention that breaks it for reading circulation, and the two uses have been served by one picture for four hundred years.

The resolution is not a better projection. It is two pictures, or one picture and a stated caveat — and the caveat now has a number attached to it, per latitude, which is the thing it did not have.

Where the model stops #

The curl is a separate question and this rung does not answer it. Vorticity is the other operator a flow map is read for, it is not preserved by an equal-area map, and which family preserves it is a different calculation. Nothing here should be read as saying an equal-area map is the right choice for a flow field in general; what it says is that an equal-area map is the only kind that preserves this property.

A drawn flow map is not a numerical field. Arrows are placed at chosen points, at a chosen gauge, and a reader integrating them by eye is doing something much rougher than a finite difference — the same objection the ellipses are a sample makes about the indicatrix field. What the measurement establishes is that the information in the picture has the defect, whether or not any particular reader extracts it.

And an interrupted map fails differently rather than worse. The closed form needs $\nabla \ln s$ to exist, and at the seam of an interrupted map it does not: the areal factor is smooth within a lobe, and two lobes meet along a cut across which nothing is differentiable. Inside each lobe of an interrupted sinusoidal the invented divergence is exactly zero, because the base is equal-area and interruption does not change that; at the cut there is no value to compute at all. So the honest description of Goode’s homolosine as a current map is that it is exact everywhere it draws the ocean and undefined along the lines where it stops — and its cuts are the ones that run through the oceans, because they were placed to keep the continents whole.

And a real current is not divergence-free. Ocean surface flows have genuine convergence at fronts and genuine divergence at upwellings, which is what the maps are drawn to show. That is what makes the invented divergence damaging rather than merely wrong: it is the same size as the signal, at the same places, and it has no signature that distinguishes it.

The generalisation #

The rule is that which projection is right depends on which derivative the reader is going to take.

This collection’s usual advice sorts by what the map preserves — angles, areas, distances — and matches that to what the reader cares about. That advice is one level too coarse for a field. A scalar field’s gradient is a direction and a magnitude, and a conformal map keeps the direction; a vector field’s divergence is a ratio of flux to area, and an equal-area map keeps it. Same reader, same data, opposite answer.

The rule generalises to any derived quantity. Ask what operator the reader will apply, then ask what that operator is made of. An operator built out of angles wants a conformal map; one built out of areas wants an equal-area one; one built out of both — vorticity, strain rate, the second derivative of a field — wants something that does not exist, and the honest response is to compute it on the sphere and map the answer rather than to map the field and compute on the page.

That last sentence is the whole practical content and it costs nothing. Computing the divergence on the ground and drawing that as a scalar field is one extra step and is immune to all of this, because a scalar quantity’s value at a point is the same number whatever page it is printed on.

Who found it, and when #

The transformation of the divergence under a change of coordinates is standard vector calculus, and it is in every treatment of fluid dynamics in curvilinear coordinates. Physical oceanography and meteorology handle it correctly as a matter of course — a numerical model integrates on a stated grid with the metric terms in, and a modeller who dropped them would find their mass not conserving within an hour of running.

Where it is not handled is in the presentation. A model computes on a sphere and its output is drawn on a page, and the drawing is where the property is lost. Nobody claims otherwise; nobody states it either, and a reader of a published current map has no way to know whether the convergence they are looking at is in the ocean or in the projection.

The one branch of the subject that has faced it squarely is plate tectonics, where the whole point of a velocity field is its derivatives — the strain rate — and where those derivatives are always computed on the sphere before anything is drawn. That discipline arrived at the habit because its numbers are compared against seismic and geodetic observations, so an error in the metric shows up against something external. Flow maps in the general case are checked against nothing.

The cheap habit, for a reader of any drawn vector field: look at the projection before believing a convergence. If the map is equal-area, the convergence is in the data. If it is not, some of it belongs to the page, the share grows towards the poles, and its size is the flow’s component along the direction in which the map’s areas are changing fastest.

Where the ladder goes next #

Eight rungs have priced what a projection does to a field drawn on it — a gradient, a contour, a watershed, a light, a second derivative and now a divergence. Every one of those takes the field as given and exact. A measured field is neither: it is a set of samples with a spacing, and the operators above are all differences, which is the one operation a spacing decides completely.